Show that sq. root of 2 to power sq. root of 2 to N converges

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Homework Help Overview

The discussion revolves around the convergence of a sequence defined by repeated exponentiation of the square root of 2. Participants are exploring the implications of the Monotone Convergence Theorem in this context.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the definition of the sequence, specifically whether it is defined as a_n+1 = a_n^sqrt(2) or a_n+1 = sqrt(2)^a_n. There is also a suggestion to compute terms to investigate convergence behavior.

Discussion Status

The discussion is currently focused on clarifying the sequence definition and its implications for convergence. Some participants have pointed out the need for precise notation and the importance of defining terms clearly to facilitate the application of the Monotone Convergence Theorem.

Contextual Notes

There is ambiguity in the sequence definition, which may affect the analysis of convergence. Participants are encouraged to clarify their definitions and assumptions before proceeding with the proof.

sigdel977
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Show that sq. root of 2 to power sq. root of 2 to ... N converges

Sq. root of 2^sq root of 2 ^ sq. of 2...N
Use monotonous convergence theorem to that it converges and determine what it converges to.
 
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Your definition of the sequence is ambiguous. Do you mean a_n+1=a_n^sqrt(2) or a_n+1=sqrt(2)^a_n? Try punching out some terms on a calculator of each. One of them doesn't converge. Then tell me what you need to prove about the sequence to use the monotone convergence theorem.
 


i ment,
(sq. rt of 2)^(sq. rt of 2)^(sq. rt of 2)^N
 


sigdel977 said:
i ment,
(sq. rt of 2)^(sq. rt of 2)^(sq. rt of 2)^N

sqrt(2)^sqrt(2)^sqrt(2) doesn't mean anything. (sqrt(2)^sqrt(2))^sqrt(2) and sqrt(2)^(sqrt(2)^sqrt(2)) are different. Use parentheses. It's easier if you define a_n+1 in terms of a_n.
 

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